Description of sllle_wg

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sllle_wg

PURPOSE

SLLLE_WG Solves the Locally Linear Embedding from weight graph

SYNOPSIS

function [Y, spectrum] = sllle_wg(G, d)

DESCRIPTION

SLLLE_WG Solves the Locally Linear Embedding from weight graph

 $ Syntax $
   - Y = sllle_wg(G, d)
   - [Y, spectrum] = sllle_wg(G, d)

 $ Arguments $
   - G:        The weights graph
   - d:        The dimension of the embedding
   - Y:        The sample coordinates in the embedded space
   - spectrum: The spectrum of the embeded space dimensions

 $ Description $
   - Y = sllle_wg(G, d) solves the locally linear embedding from
     a given weight graph. G should be a graph of n nodes, where
     the edge value from i-th node to j-th node, means the weights
     on the i-th sample in constructing the j-th sample, (or the
     construction of i-th sample to the j-th sample). 
     Y is solved by taking the eigenvectors of (I - W)(I - W)^T, 
     corresponding to the (d+1) smallest eigenvalues, and discarding 
     the smallest one.  In our output, the dimension is sorted in the 
     ascending order of eigenvalues.

   - [Y, spectrum] = sllle_wg(G, d) also returns the spectrum of the
     corresponding dimensions, a column vector of the corresponding
     eigenvalues.

 $ Remarks $
   - The dimension d should be strictly less than n.

 $ History $
   - Created by Dahua Lin, on Sep 11st, 2006

CROSS-REFERENCE INFORMATION

This function calls:

• slsymeig SLSYMEIG Compute the eigenvalues and eigenvectors for symmetric matrix
• sladjmat SLADJMAT Constructs the adjacency matrix representation of a graph
• slgraphinfo SLGRAPHINFO Extracts basic information of a given graph representation
• raise_lackinput RAISE_LACKINPUT Raises an error indicating lack of input argument

This function is called by:

• sllle SLLLE Performs Locally Linear Embedding

SOURCE CODE

0001 function [Y, spectrum] = sllle_wg(G, d)
0002 %SLLLE_WG Solves the Locally Linear Embedding from weight graph
0003 %
0004 % $ Syntax $
0005 %   - Y = sllle_wg(G, d)
0006 %   - [Y, spectrum] = sllle_wg(G, d)
0007 %
0008 % $ Arguments $
0009 %   - G:        The weights graph
0010 %   - d:        The dimension of the embedding
0011 %   - Y:        The sample coordinates in the embedded space
0012 %   - spectrum: The spectrum of the embeded space dimensions
0013 %
0014 % $ Description $
0015 %   - Y = sllle_wg(G, d) solves the locally linear embedding from
0016 %     a given weight graph. G should be a graph of n nodes, where
0017 %     the edge value from i-th node to j-th node, means the weights
0018 %     on the i-th sample in constructing the j-th sample, (or the
0019 %     construction of i-th sample to the j-th sample).
0020 %     Y is solved by taking the eigenvectors of (I - W)(I - W)^T,
0021 %     corresponding to the (d+1) smallest eigenvalues, and discarding
0022 %     the smallest one.  In our output, the dimension is sorted in the
0023 %     ascending order of eigenvalues.
0024 %
0025 %   - [Y, spectrum] = sllle_wg(G, d) also returns the spectrum of the
0026 %     corresponding dimensions, a column vector of the corresponding
0027 %     eigenvalues.
0028 %
0029 % $ Remarks $
0030 %   - The dimension d should be strictly less than n.
0031 %
0032 % $ History $
0033 %   - Created by Dahua Lin, on Sep 11st, 2006
0034 %
0035 
0036 %% parse and verify input
0037 
0038 if nargin < 2
0039     raise_lackinput('sllle_wg', 2);
0040 end
0041 
0042 gi = slgraphinfo(G, {'square'});
0043 if ~gi.valued
0044     error('sltoolbox:invalidarg', 'The graph G should be a valued graph');
0045 end
0046 if isnumeric(G)
0047     W = G;
0048 else
0049     W = sladjmat(G, 'sparse', true);
0050 end
0051 
0052 n = gi.n;
0053 if d >= n
0054     error('sltoolbox:invalidarg', 'd should be strictly less than n');
0055 end
0056 
0057 %% compute
0058 
0059 if issparse(W)
0060     M = speye(n) - W;
0061 else
0062     M = eye(n) - W;
0063 end
0064 clear W;
0065 M = M * M';
0066 
0067 [spectrum, Y] = slsymeig(M, d+1, 'ascend');
0068 clear M;
0069 spectrum = spectrum(2:d+1);
0070 Y = Y(:, 2:d+1);
0071 Y = Y';
0072 
0073

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